Another Obedient Ray

Two mirrors, A and B, are joined at a fixed angle at point O, and
a ray of light is shone into the angle between them. It first
strikes one mirror at a point X. When the ray re-emerges, the last
point it strikes is point Y on one mirror.

If the incident ray is parallel to mirror A, and the exit ray is
parallel to mirror B, find the possible angles for AOB and the
possible relationships between OX and OY.

Find the requirements on the angle of incidence, with respect to
angle AOB, such that OX=OY.